The minimal degree of plane models of double covers of smooth curves

TL;DR

This paper investigates bounds on the minimal degree of plane models for double covers of smooth curves, relating these bounds to gonality and genus, and provides examples where bounds are sharp.

Contribution

It establishes new upper and lower bounds for the minimal degree of plane models of double covers based on gonality and genus, including cases where bounds coincide.

Findings

Bounds are tight when the base curve is hyperelliptic.

Examples demonstrate the minimal degree can achieve the lower bound.

Results extend understanding of plane models of double covers.

Abstract

If $X$ is a smooth curve such that the minimal degree of its plane models is not too small compared with its genus, then $X$ has been known to be a double cover of another smooth curve $Y$ under some mild condition on the genera. However there are no results yet for the minimal degrees of plane models of double covers except some special cases. In this paper, we give upper and lower bounds for the minimal degree of plane models of the double cover $X$ in terms of the gonality of the base curve $Y$ and the genera of $X$ and $Y$. In particular, the upper bound equals to the lower bound in case $Y$ is hyperelliptic. We give an example of a double cover which has plane models of degree equal to the lower bound.